I've been thinking a lot about this question over the past few days (it's disrupted other planned activity, for which I blame Sunstone, but luckily foremost among what I'd planned to finish was in the way of commentary over some reading material I now have a scapegoat for). Neuroscientists tend to have a reputation for being the most hard-core reductivists around, as many of them come from backgrounds in psychology and approach the biological sciences (of which only a minority go beyond the very basics) as a necessary evil that need only be understood to the point at which psychological phenomenon can be descriptively reduced to biological processes. However, there are a lot of exceptions, although a great many of these aren't neuroscientists so much as they are specialists from other fields who have used their expertise in tandem with neurologists, psychiatrists, neuroscientists, and others whose study is centered on the brain. Then there are those of us who are interested in closing the gap between computational neuroscience and cognitive neuroscience, which is to say constructing explanatory models of cognitive processes through neurophysiological processes.
I started down my path to insanity (well, another kind of insanity than that I already had) by trying to show that, at the very least, we could rule out quantum theories of consciousness (and not via reliance on the quantum-to-classical transition). This was a mistake, as on the road to quantum physics lies madness. I usually describe my position as being that of a physicalist, but one of the problems with modern physics is that many physical systems are mathematical entities. It's not just that the fundamental constituents of all matter are (in the standard view) probability functions. It's not even the ontological indeterminacy. It's the ways in which the classical world we experience can be recovered through our forcing reality to exist as one thing vs. another. This is a central epistemic and philosophical basis (sometimes explicitly stated, as with Stapp's quantum consciousness theory) for quantum theories of mind: we seems to decide to do things in ways that nothing in classical physics can explain but that is built into quantum theory. To me, this is only a good starting point if one can use it as a foundation for identifying how quantum processes give rise to or even contribute to what little we know of the physics of consciousness, conceptual processing, & cognition in general. It hasn't, and until someone supporting such theories can offer this I think it is actually preventing (as quantum physics has in general) us from looking anew at "classical physics".
It is often somehow assumed that classical physics is only incomplete when it comes to the atomic/sub-atomic scale (or mechanics at scales astrophysicists deal with if one doesn't view special relativity as classical physics). That is, it's sort of tacitly taken for granted that if we don't need quantum physics or relativistic physics we can rely on classical physics. In reality, classical physics at its best wasn't a very good tool to understand most physical phenomena, but was virtually bankrupt at explaining the dynamics of living systems. And, while modern physics & chemistry have made tremendous strides in our ability to model, explain, predict, and in general understand even complex systems that aren't living, the glaring failure of similar progress in biology has motivated a great many to propose that we don't understand the classical realm as well as we thought.
Classical physics arose out of natural philosophy as classical mechanics in order to explain why inanimate things moved the way they did. It continued to do mainly this right up through the origins of quantum physics. Take a system as complex as you'd like, such as the climate, and despite all the nonlinearities, highly complex interactions between and among "networks", etc., it's still all about how forces act on inanimate objects to make them move in particular ways. Living systems (even single-celled organisms) are qualitatively different. They are animate, for one thing, and classical physics was developed to explain the inanimate. Our models of their dynamics involve constant appeals to processes that we use to explain the dynamics of the "parts" of the system, only somehow these parts are also causing the functional processes.
Concepts, however they relate to the physical brain (and not just those of humans), are perhaps the greatest challenge to the fundamental ideas that drove the development of classical physics for centuries. Concepts involved in awareness, the sense of "I/me" that is self-awareness/consciousness, even those of desires appear fundamentally different kinds of "forces" than those that motivated most of physics up-to and including today: ethereal, seemingly non-corporeal abstractions that some how act upon a physical system causally. Moreover, the brain is in some ways the most complex system known (trivially speaking, it clearly must not be as the body contains the brain and much else, and therefore is necessarily more complex). We can not only model to a high degree of accuracy the kind of "learning" or reactions to environment that most living systems are capable of, but many of the methods used in AI, soft computing, computational intelligence, etc., are based off of how living systems of this type "learn" (non-conceptually). We are not remotely close to understanding or creating models of living systems capable of conceptual processing. So we have a physics that is particularly inadequate when it comes to understanding even the simplest life that we wish to use to understand something that runs counter to the foundations for classical physics and in most ways physics itself. Meanwhile, just what it means to be "physical" has becoming increasingly less clear.
When the most successful theory of all time tells you that a thing can be both A & ~A and that everything is made up of nonlocal, indiscrete components (there are no "particles", just things that are wave-like but whose wave-like properties become infinitesimal as the the size of the system exceeds its de Broglie wavelength), there's reason to think deeply about what it means to describe something as "material" or "physical. And as this theory didn't develop like the (mostly) progressive successes of classical physics but was born out of the catastrophic realization that the entire framework of physics was fundamentally flawed, are we really so justified in relying on that framework except where it failed so spectacularly before?